cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A359136 Primes such that there is a nontrivial permutation which when applied to the digits produces a prime (Version 1).

Original entry on oeis.org

11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 419, 421
Offset: 1

Views

Author

N. J. A. Sloane and Allan C. Wechsler, Jan 22 2023

Keywords

Comments

A prime p with decimal expansion p = d_1 d_2 ... d_m is in this sequence iff there is a non-identity permutation pi in S_m such that q = d_pi(1) d_pi(2) ... d_pi(m) is also a prime. The prime q may or may not be equal to p. Leading zeros are permitted in q.
One must be careful when using the phrase "nontrivial permutation of the digits". When the first and third digits of 101 are exchanged, this is applying the nontrivial permutation (1,3) in S_3 to the digits, leaving the number itself unchanged. One should specify whether it is the permutation that is nontrivial, or its action on what is being permuted. In this sequence and A359137, we mean that the permutation itself is nontrivial.
There are only 34 primes not in this sequence, the greatest of which is 5849. - Andrew Howroyd, Jan 22 2023

Links

Crossrefs

Many OEIS entries are subsequences (possibly after omitting 2, 3, 5, and 7): A007500, A055387, A061461, A069706, A090933, A225035.
Cf. A359137, A359138, A359139.

Programs

  • PARI
    isok(n)={my(v=vecsort(digits(n))); if(#Set(v)<#v, 1, forperm(v, u, my(t=fromdigits(Vec(u))); if(isprime(t) && t<>n, return(1))); 0)} \\ Andrew Howroyd, Jan 22 2023
  • Python
    from sympy import isprime
from itertools import permutations as P
def ok(n):
    if not isprime(n): return False
    if len(s:=str(n)) > len(set(s)): return True
    return any(isprime(t) for t in (int("".join(p)) for p in P(s)) if t!=n)
print([k for k in range(422) if ok(k)]) # Michael S. Branicky, Jan 23 2023

Extensions

More than the usual number of terms are shown in order to distinguish this from neighboring sequences.
Incorrect terms removed by Andrew Howroyd, Jan 22 2023

A085298 a(n) is the smallest exponent x such that prime(n)^x when reversed is a prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 8, 7, 1, 1, 2, 5, 15, 10, 12, 4, 39, 1, 1, 1, 11, 2, 1, 1, 10, 1, 23, 1, 5, 1, 243, 2, 1, 1, 1, 23, 1, 34, 1, 1, 1, 2, 58, 1, 3, 9, 166, 17, 68, 8, 8, 3, 7, 5, 5, 2, 2, 2, 61, 11, 97, 1, 1, 10, 2, 1, 1, 41, 1, 1, 66, 1, 5, 1, 1, 2, 2, 8, 40, 2, 8, 19, 2, 2, 723
Offset: 1

Views

Author

Labos Elemer, Jun 24 2003

Keywords

Comments

It is conjectured that for every n such exponent exists.

Examples

			a(n)=1 means that rev(prime(n)) is prime i.e. prime(n) is in A007500;
a(n)=2 means that rev(prime(n)^2) is prime but rev(prime(n)) is not, like n=8:p=19 and 91 is not a prime but rev[19^2]=rev[361]=163 is a prime;
For n, the first k exponent providing rev(prime(n)^k) prime can be quite large, like at n=87: rev(p(87)^723)=rev(449^723) is the first [probably] prime has 1918 decimal digits: 948......573.

Links

Crossrefs

Cf. A000040, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

Programs

  • Maple
    a:= proc(n) local k, p; p:= ithprime(n); for k while not isprime((s->
      parse(cat(seq(s[-i], i=1..length(s)))))(""||(p^k))) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 04 2019
  • Mathematica
    a[n_] := Block[{k = 1}, While[! PrimeQ@ FromDigits@ Reverse@ IntegerDigits[ Prime[n]^k], k++]; k]; Array[a, 87] (* Giovanni Resta, Sep 04 2019 *)
  • PARI
    a(n) = {my(x=1, p=prime(n)); while (!ispseudoprime(fromdigits(Vecrev(digits(p^x)))), x++); x;} \\ Michel Marcus, Sep 04 2019

Formula

a(n) = Min{x; reversed(prime(n)^x) is a prime}.

A085300 a(n) is the least prime x such that when reversed it is a power of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 163, 18258901387, 90367894271, 13, 73, 1861, 344800741, 34351783286302805384336021, 940315563074788471, 1886172359328147919771, 14854831
Offset: 1

Views

Author

Labos Elemer, Jun 24 2003

Keywords

Comments

A006567 (after rearranging terms) and A002385 are subsequences. - Chai Wah Wu, Jun 02 2016

Examples

			a(14)=344800741 means that 147008443=43^5=p(14)^5, where 5 is the smallest such exponent;
a(19) has 82 decimal digits and if reversed equals 39th power of p(19)=67.

Links

Crossrefs

Cf. A085298, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

Programs

  • Python
    from sympy import prime, isprime
def A085300(n):
    p = prime(n)
    q = p
    while True:
        m = int(str(q)[::-1])
        if isprime(m):
            return(m)
        q *= p # Chai Wah Wu, Jun 02 2016

A085299 a(n) is the smallest number x such that A085298[x]=n, or 0 if no such number exists.

Original entry on oeis.org

1, 8, 47, 18, 14, 89, 10, 9, 48, 16, 23, 17, 168, 268, 15, 661, 50, 380, 84, 116, 360, 245, 29, 144, 345, 227, 785, 261, 148, 235, 691, 658, 638, 40, 1023, 674, 1529, 210, 19, 81, 181, 428, 170, 1130, 2322, 406, 600, 373, 958, 217
Offset: 1

Views

Author

Labos Elemer, Jun 24 2003

Keywords

Examples

			a(13) = 168 means that 13 is the smallest exponent such that reversed[p(168)^13] = reversed[997^13] = 776831144302925059735912605306533496169
is prime if read in this direction and 13th prime-power if read backwards.

Crossrefs

Cf. A085298, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

A234901 Primes which remain prime when the digits are rotated once to the right.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 173, 197, 199, 271, 277, 311, 313, 337, 373, 379, 397, 419, 479, 491, 571, 577, 593, 617, 631, 673, 719, 733, 811, 839, 877, 911, 919, 971, 977, 991, 1031, 1039, 1091, 1093, 1097, 1171, 1193, 1213
Offset: 1

Views

Author

Colin Barker, Jan 01 2014

Keywords

Examples

			The prime 1097 is in the list because 7109 is also prime.

Links

Crossrefs

Cf. A061461, A227916, A227919.

Programs

  • Magma
    r:=func; [p: p in PrimesUpTo(1300) | IsPrime(r(p))]; // Bruno Berselli, Jul 04 2014
  • Mathematica
    Select[Prime[Range[200]],PrimeQ[FromDigits[RotateRight[IntegerDigits[#]]]]&] (* Harvey P. Dale, May 05 2022 *)
  • PARI
    rotr(a) = if(a<10, a, eval(Str(a%10, a\10)))
s=[]; forprime(n=2, 2000, if(isprime(rotr(n)), s=concat(s, n))); s

A359978 Prime numbers whose decimal representation can be split into two nonempty parts without leading 0, say x and y (in that order), and such that the concatenation of y and x yields a different prime number.

Original entry on oeis.org

13, 17, 31, 37, 71, 73, 79, 97, 113, 127, 131, 149, 157, 163, 173, 181, 191, 197, 199, 271, 277, 311, 313, 317, 331, 337, 359, 367, 373, 379, 397, 419, 479, 491, 571, 577, 593, 617, 631, 673, 719, 727, 733, 739, 757, 761, 787, 797, 811, 839, 877, 911, 919, 937
Offset: 1

Views

Author

Rémy Sigrist, Jan 20 2023

Keywords

Examples

			The first terms, alongside the corresponding other prime numbers, are:
  n   a(n)  Other prime numbers
  --  ----  -------------------
   1    13  {31}
   2    17  {71}
   3    31  {13}
   4    37  {73}
   5    71  {17}
   6    73  {37}
   7    79  {97}
   8    97  {79}
   9   113  {131, 311}
  10   127  {271}
  11   131  {113, 311}
  12   149  {491}

Links

Crossrefs

Cf. A061461, A225035, A246944.

Programs

  • PARI
    See Links section.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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